Find the number of positive integers $n$ that satisfy
\[(n - 1)(n - 3)(n - 5) \dotsm (n - 97) < 0.\]
Explanation: We can count that there are 49 factors in the given product.  For $n < 1,$ all the factors are negative, so the product is negative.

Then for $1 < n < 3,$ the factor $n - 1$ changes sign, and the product becomes positive.  For $3 < n < 5,$ the product changes sign again, and the product becomes negative, so the inequality holds for $n = 4.$

Continuing in this way, we see that the inequality holds for $n = 4,$ 8, 16, $\dots,$ 96.  For $n > 97,$ all the factors are positive, so the total number of such integers is $\boxed{24}.$